5 p x k p x k 05 gaussian approximation with

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Unformatted text preview: entered at zero), shrinking it by a factor σ horizontally and stretching it by a factor σ vertically. Let’s check that the N (µ, σ 2 ) density indeed integrates to one, has mean µ, and variance σ 2 . To show that the normal density integrates to one, it suffices to check that the standard normal density integrates to one, because the density for general µ and σ 2 is obtained from the standard normal 2 ∞ pdf by the scaling rule, which preserves the total integral of the density. Let I = −∞ e−u /2 du. Then, switching to polar coordinates, we have: ∞ 2 /2 e−u I2 = −∞ ∞ ∞ −∞ 2π ∞ du −∞ ∞ 2 +v 2 )/2 e−(u e−r = 0 ∞ = 2π 2 /2 dv −∞ = 0 e −v e −r 0 = −2πe−r 2 /2 2 /2 2 /2 dudv rdrdθ rdr ∞ = 2π. 0 1 http://www.stat.tamu.edu/∼west/applets/normaldemo.html for example. 3.6. LINEAR SCALING OF PDFS AND THE GAUSSIAN DISTRIBUTION 91 √ Therefore, I = 2π, which means the standard normal density integrates to one, as claimed. The fact that µ is the mean of the N (µ, σ 2 ) density follows from the fact that...
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